Abstract

Alternative Hedging Strategies in the Presence of Jump Risks

M. Schulmerich and S. Trautmann

Universität Mainz

In a complete financial market every contingent claim can be hedged perfectly. In an incomplete market the writer of a contingent claim can avoid any shortfall risk by following a so-called superhedging strategy. In both situations the corresponding strategies may require more initial hedging capital than the hedger is willing or able to invest. This paper examines various recent hedging approaches to cope with such an situation when the stock price follows a Poisson jump diffusion process with lognormally distributed jump sizes. First of all we apply the Expected Shortfall-Hedging} approach as pioneered by Föllmer/Leukert (1998,1999), Cvitanic (1998) and Cvitanic/Karatzas (1998) to the jump diffusion setting and provide a sufficient condition for the optimality of the trading strategy. Second, we provide a similar condition when minimizing the shortfall probability according to Föllmer/Leukert's (1999) quantile-hedging approach. Furthermore, we compare Schweizer's Locally Risk Minimizing (LRM) approach especially with Merton's (1976) delta-hedging strategy. While in the latter strategy diffusion risk is perfectly hedged while jump risk remains unhedged, the LRM-strategy hedges both diffusion risk as well as jump risk partly. We relate the LRM approach to the so-called locally variance-minimizing (LVM) hedging strategy in Bates' (1991) systematic jump risk model. By numerical analysis we find that the LRM and LVM hedge ratios are less sensitive to changes in the stock price than delta hedging strategies in the models of Merton, Black/Scholes, and Bates. If the expected jump size is significantly different from zero and positive (negative), then the LRM and LVM hedge ratios are substantially larger (smaller) than e.g. Merton's for out-of-the-money (in-the-money) calls. Moreover, the worst case behaviour of the LRM strategy and LVM strategy are substantially better: The 99\%-, 95\%-, as well as the 90\%-quantile of the total hedging costs are significantly lower than for the alternative strategies.